An electronic device uses an aluminum plate of cross-section 4"x4" to take a pure bending moment of 13,000 lb-in. The factor of safety is two. Using the properties of aluminum given in Table 3.4, find the thickness of the plate. The designer wants to at least halve the thickness of the plate to make room for additional hardware on the electronic device. The choices include unidirectional laminates of Graphite/ Epoxy, Glass/Epoxy or their combination (hybrid laminates). The ply thickness is 0.125 mm (0.0049213"). Design a plate with the lowest cost if the manufacturing cost per ply of Graphite/Epoxy and Glass/Epoxy is 10 and 4 units, respectively.

SOLUTION: The maximum normal stress in a plate under bending is given by

Stress = M t/(2 I)
where
M = bending moment (lb-in)
t = thickness of plate (in)
I = second moment of area (in^4)

For a rectangular cross-section, the second moment of area is
I = b t^3/12
where
b = width of plate (in).

Using a factor of safety, F = 2 and given b = 4 inches, then the thickness of the plate using the Maximum Stress criterion is
t= 0.9871 in

Now the designer wants to replace the 0.9871 inches thick aluminum plate by a plate of maximum thickness 0.4935 inches (half that of aluminum) made of laminated composites. The bending moment per unit width is 3,250 lb-in.

Using the factor of safety of two, the plate is designed to take a bending moment per unit width of 6250 lb.

The simplest choices are to replace the aluminum plate by an all Graphite/Epoxy laminate or an all Glass/Epoxy laminate. Using PROMAL program, the strength ratio for a single 0 degree ply for the above load for Glass/Epoxy ply is

SR = 5.4942 E-5

Since the bending moment per unit width is inversely proportional to the square of the thickness of the plate, the minimum number of plies required would be 135 which gives the thickness of the all Glass/epoxy laminate as

t = 135 x 0.0049213
= 0.664 in.

The thickness of an all Glass/Epoxy laminate is more than 0.4935 inches and is hence not acceptable. Similarly for an all Graphite/Epoxy a laminate, the minimum number of plies required would be
n = 87 plies
which gives the thickness of the plate as
t = 87 x 0.0049213
= 0.42815 in.
The thickness of an all Graphite/Epoxy laminate is less than 0.4935 inches and is hence acceptable.

Even if an all graphite/epoxy laminate is acceptable, Graphite/Epoxy is 2.5 times costlier than a Glass/Epoxy, one would suggest the use of a hybrid laminate. The question which arises now is in what sequence the plies should be stacked. In a plate under a bending moment, the magnitude of ply stresses is maximum on the top and bottom face. Since the longitudinal tensile and compressive strengths are larger in the Graphite/Epoxy ply then in a Glass/Epoxy ply, one would put them as the facing material and the Glass/Epoxy in the core. The maximum number of plies allowed in the hybrid laminate is
= 0.4936/0.0049213
= 100 plies
Several combinations of 100 ply symmetric hybrid laminates were subjected to the applied bending moment. Minimum strength ratios in each laminate stacking sequence were found. If the strength ratios are greater than one, the cost of the stacking sequence was determined. The hybrid laminate with the lowest cost is [16 plies of Graphite/Epoxy, 68 plies of Glass/Epoxy and 16 plies of Graphite/Epoxy].

The above example dictated the use of unidirectional laminates. How will the design change if there were multiple loads? Examples of multiple loads include a leaf spring subjected to bending moment as well as torsion, a thin pressure vessel subjected to an internal pressure to yield a biaxial state of stress. In such cases, one may have a choice not only of material systems or their combination but that of orientation of plies as well. Since there can be infinite combinations of angle plies, attention may be focussed on angle plies of 0, 90, 45 and -45 angle plies and their combination thereof. This reduces the possibilities to a finite number for a limited number of material systems, but still the number can be quite large to handle.

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